This part provides a background on the theory of persistent homology. Homology \(H_k(X)\)  is an essential invariant in algebraic topology, which captures the information of the \(k\) -dimensional holes (connected components, loops, cavities) in a topological space \(X\) . For example, a connected component in a graph is a zero-dimensional hole, whereas a graph loop is a 1-dimensional hole. Persistent homology is a way to use this invariant to keep track of the changes in a controlled topological space sequence induced by the original space \(X\) . For basic background on persistent homology, see [1]}, [2]}.
